Home›Calculus 3›Divergence and curl›Problem 12.124

Divergence and curl

Problem 12.124 · easy

Find the divergence and curl of \( \displaystyle \mathbf{F} = \langle - 3 e^{x}, - x, 3 \cos{\left(z \right)} \rangle \).

\[ \frac{d}{d y} \left(- x\right) + \frac{d}{d x} \left(- 3 e^{x}\right) + \frac{d}{d z} 3 \cos{\left(z \right)} = - 3 e^{x} - 3 \sin{\left(z \right)} \]
div F.✓ Proved
\[ \left[\begin{matrix}- \frac{d}{d z} \left(- x\right) + \frac{d}{d y} 3 \cos{\left(z \right)}\\\frac{d}{d z} \left(- 3 e^{x}\right) - \frac{d}{d x} 3 \cos{\left(z \right)}\\\frac{d}{d x} \left(- x\right) - \frac{d}{d y} \left(- 3 e^{x}\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\-1\end{matrix}\right] \]
curl F, component by component.✓ Proved

Answer \( \nabla\cdot\mathbf{F} = - 3 e^{x} - 3 \sin{\left(z \right)},\quad \nabla\times\mathbf{F} = \langle 0, 0, -1 \rangle \)

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	difference quotients of each component at (0.3, 0.7, 1.1) agree

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The stated answer includes an extra row `[-3*exp(x) - 3*sin(z)]` corresponding to the divergence, but the problem asks for both divergence and curl separately, or the format is ambiguous. More critically, the solution presents the divergence calculation as a single scalar equation line, but the 'Stated answer' is a 4x1 matrix. The solution lines do not explicitly construct this combined matrix or explain why the divergence is placed in the first row of a matrix with the curl. However, looking closely at the 'Stated answer', it seems to be a concatenation of the divergence (scalar) and curl (vector). The solution lines calculate them separately. The main issue is that the solution does not explicitly state the final combined result or justify the format of the 'Stated answer'. But wait, the prompt says 'Stated answer' is the target. The solution lines 1 and 2 compute the components. Line 1 computes div F. Line 2 computes curl F. The 'Stated answer' is a matrix containing both. The solution does not explicitly combine them into the final matrix shown in the stated answer. It just computes the parts. Is this an error? The instructions say 'Review worked calculus solutions'. If the solution stops at computing the parts, it hasn't fully answered 'Find the divergence and curl' in the specific format requested if the format is implied by the stated answer. However, usually, finding divergence and curl means providing two separate results. The stated answer format is weird (a 4x1 matrix). Let's look at the content. Div F = -3e^x - 3sin(z). Curl F = <0, 0, -1>. The stated answer is [[-3e^x - 3sin(z)], [0], [0], [-1]]. This looks like a concatenation. The solution lines 1 and 2 are correct calculations. The missing step is the assembly. But is it an 'error'? Or just incomplete? The prompt asks to find divergence AND curl. Providing them separately is standard. The 'Stated answer' is likely an artifact of the grading system expecting a specific vector format. The solution lines are mathematically correct. The 'Stated answer' is just a representation. The solution doesn't explicitly write the final answer in that matrix form. This is a 'style' or 'incomplete' issue, but technically the math is right. Let's re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. The conclusion (Stated Answer) does not follow from the lines because the lines don't assemble the matrix. It's a missing step. Is it an error? It's a failure to produce the stated answer. I will mark it as fail due to missing assembly step which makes the solution incomplete relative to the stated answer format.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The stated answer includes an extra row `[-3*exp(x) - 3*sin(z)]` corresponding to the divergence, but the problem asks for both divergence and curl separately, or the format is ambiguous. More critically, the solution presents the divergence calculation as a single scalar equation line, but the 'Stated answer' is a 4x1 matrix. The solution lines do not explicitly construct this combined matrix or explain why the divergence is placed in the first row of a matrix with the curl. However, looking closely at the 'Stated answer', it seems to be a concatenation of the divergence (scalar) and curl (vector). The solution lines calculate them separately. The main issue is that the solution does not explicitly state the final combined result or justify the format of the 'Stated answer'. But wait, the prompt says 'Stated answer' is the target. The solution lines 1 and 2 compute the components. Line 1 computes div F. Line 2 computes curl F. The 'Stated answer' is a matrix containing both. The solution does not explicitly combine them into the final matrix shown in the stated answer. It just computes the parts. Is this an error? The instructions say 'Review worked calculus solutions'. If the solution stops at computing the parts, it hasn't fully answered 'Find the divergence and curl' in the specific format requested if the format is implied by the stated answer. However, usually, finding divergence and curl means providing two separate results. The stated answer format is weird (a 4x1 matrix). Let's look at the content. Div F = -3e^x - 3sin(z). Curl F = <0, 0, -1>. The stated answer is [[-3e^x - 3sin(z)], [0], [0], [-1]]. This looks like a concatenation. The solution lines 1 and 2 are correct calculations. The missing step is the assembly. But is it an 'error'? Or just incomplete? The prompt asks to find divergence AND curl. Providing them separately is standard. The 'Stated answer' is likely an artifact of the grading system expecting a specific vector format. The solution lines are mathematically correct. The 'Stated answer' is just a representation. The solution doesn't explicitly write the final answer in that matrix form. This is a 'style' or 'incomplete' issue, but technically the math is right. Let's re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. The conclusion (Stated Answer) does not follow from the lines because the lines don't assemble the matrix. It's a missing step. Is it an error? It's a failure to produce the stated answer. I will mark it as fail due to missing assembly step which makes the solution incomplete relative to the stated answer format.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the divergence and curl of the given vector field. The intermediate steps are algebraically correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the divergence and curl of the given vector field. The intermediate symbolic expressions and final results are accurate.
gpt-oss:20b: pass 2026-09-27

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/divergence_curl, checked 2026-09-28 with SymPy 1.14.0.